Sun illuminates tall objects
This could be happening because there is no geometry to block the light.
At 12AM, the sun is below the horizon, so the Earth should cast a large shadow on the city. In your case, however, there is no geometry to block this light, and so the sun is able to shine on your tower from below.
You could create a huge plane at z=0 to emulate the Earth's shadow. Alternatively, as you suggested, you can animate the sun's strength to make it dark at night.